Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

where gμ = Tμ g.4 The Davidenko’s method fails at turning points (e.g. saddle-node bifurcations) because of the singularity of gy . More details on homotopy techniques can be found in [276].

Equations (5.60) is equivalent to a set of ODE, where the integration variable is μ. It is relevant to note the similarity of the continuous Newton’s method (4.70) and (5.60). In fact, let us define the function h(y, t) as follows:

where e represents the natural exponential. Then, di erentiating h, one has:

Equation (5.63) shows that t can be viewed as the continuation parameter for the function h(y, t). As for the Davidenko’s method, the continuous Newton’s method fails at turning points where the power flow Jacobian matrix is singular.

The main di erence between (5.60) and (5.63) is that μ is an internal parameter (i.e., the homotopy is forced) while t is an external one (i.e., the homotopy is free-running). Thus, only the final equilibrium point of (5.63) is physically relevant, while the values of y in intermediate iterations lack of physical meaning.

5.4.5N-1 Contingency Analysis

On the use of the CPF analysis there is sometime some confusion. The most frequent critic that is moved to the CPF model is that the path used for increasing load powers cannot be known a priori. Another critic is about the use of a scalar parameter μ that increases all load and generator powers.

Actually, these critics are not justified since the basic assumption of continuation power flow analysis is that the load increase is virtual. In other words, the information provided by the CPF is how distant from the point of collapse is the current loading condition. Thus, the loading level μ is not a

4Equation (5.60) can be also obtained from (4.70). As a matter of fact, di erentiating (4.68) and imposing dy˙ = 0 leads to:

where it is implicitly assumed that gy does not depend on μ, as discussed in Section 5.2.

real load increase, but rather a measure of the security margin of the current loading condition.

The usefulness of the CPF analysis is clear if applied to contingency analysis. System operators have to ensure that for each credible contingency (e.g., line or generator outage) the system is able to supply the current load. This is also referred as N-1 contingency criterion. The CPF provides a tool to evaluate the impact of each contingency in terms of the maximum loading level μmax. If a contingency leads to a μmax < 1, then the contingency is not feasible.

System operators generally fix a minimum loading margin to ensure that the current loading condition has a minimum distance to the point of collapse. Thus, for each contingency, μmax ≥ μsm must hold, where μsm is the required security margin. For example, if μsm = 1.1, the system has to ensure a 10% loading margin. If a certain contingency is characterized by μmax < μsm, then that contingency is classified as critical and corrective actions are recommended.

Example 5.5 N-1 Contingency Analysis for the IEEE 14-Bus System

Figure 5.12 and Table 5.1 illustrate the N-1 contingency analysis for the IEEE 14-bus test system. Results are obtained considering only generator reactive power limits. The table shows the maximum loading level μmax correspondent

Fig. 5.12 Nose curves for the IEEE 14-bus system considering a variety of line outages

to the outage of each line of the network. The outage is considered feasible if the maximum loading condition associated to line outage is μmax > 1 (μ = 1 is the base case loading condition). According to this condition, only line 1-2 outage is unfeasible. Moreover, assuming for the sake of example that the required security margin is μsm = 1.3 (i.e., 30% loading margin), the line 2-3 outage is critical.

5.5Summary

This section summarizes most relevant concepts related to continuation power flow analysis.

Maximum loading condition: The calculation of the maximum loading condition of a power system can be formulated as the problem of computing the bifurcation points of classical power flow equations. Although bifurcation theory applies to dynamical systems, the use of static power flow equations is justified by assuming slow load variations and fast dynamics of power flow variables.

Point of collapse and bifurcation points: The only bifurcations of interest in power flow analysis are those that lead to a point of collapse, i.e., the

maximum loading level beyond which power flow equations have no solution. These are the saddle-node and the limit-induced bifurcations. The saddle-node bifurcation is associated with the transmission system capacity and is intrinsic of the quadratic form of power flow equations. The limit-induced bifurcation refers to the reactive power reserve of synchronous generators. While the saddle-node bifurcation is always a point of collapse, limit-induced bifurcations can be critical or non-critical. Only critical limit-induced bifurcations are points of collapse.

Direct methods: Direct methods allows calculating directly the maximum loading condition of the system. The simplest direct methods are formulated as a set of nonlinear equations that include power flow equations and other constraints aimed to impose the conditions of either the saddle-node or the limited-induced bifurcation. Such direct methods are di cult to solve and have no practical interest. Another class of direct methods is based on nonlinear programming techniques. These lead to the formulation of the maximum loading condition as an optimal power flow problem. Due to its several implications that go far beyond the determination of the maximum loading condition, the entire Chapter 6 is dedicated the discussion of the optimal power flow analysis.

Homotopy methods: Homotopy methods are a class of algorithms that allow e ciently and reliably solving the problem of calculating the maximum loading condition of power flow equations. Homotopy methods consist in defining a homotopy map and a continuation equation whose Jacobian matrix is not singular at bifurcation points, hence the numerical robustness. In particular, the continuation power flow analysis is a forced homotopy method and consists in a predictor and in a corrector step. The predictor can be obtained through a tangent vector or the secant, whereas the corrector can be obtained through a perpendicular intersection or a local parametrization.

N-1 contingency analysis: The maximum loading condition can be considered as a measure of the distance to the collapse of the current operating point and not the other way round, i.e., the amount of load that the system can feed before collapsing. Thus, the most relevant application of the CPF technique is the N-1 contingency analysis. Given a set of contingencies (e.g., line outages), the CPF analysis provides the maximum loading level corresponding to each contingency. If a contingency is characterized by a maximum lading level lower than the current operating condition, that contingency is unfeasible. If a contingency is characterized by a maximum lading level lower than a given margin (typically fixed by the system operator), that contingency is critical. In both cases, the system operator has to take corrective actions to improve system security.

3

−pL3

Fig. 6.1 3-bus system

As a simple example of optimization problem, consider the classical problem of generator cost minimization. Since this 3-bus system is loss-less, one has:

where cG1 and cG2 are generator cost functions and can be assumed quadratic. For example, cG1(pG1) is:

A similar expression holds for cG2(pG2).

The solution of (6.1)-(6.2) is straightforward if one defines the Lagrangian function and imposes the Karush-Kuhn-Tacker’s (KKT) optimality conditions. The Lagrangian function is:

Hence the KKT condition are:

where ρ is the dual variable or Lagrangian multiplier associated with the power balance equation (6.2). Thus, the solution is:

The dimension of the dual variable ρ is a cost per power unit and per hour. For this reason, ρ is also called marginal cost of the system. In a restructured power system, generator costs have become “secret” and only o ers are known. However, as it often happens, this is only a new name for the same mathematical quantity. If one assumes that functions cG1(pG1) and cG2(pG2) are o ers, the dual variable ρ takes the meaning of market clearing price, i.e., the amount that each generator of the system has to be paid for each unit of produced power. In any case, the solution procedure of (6.1)-(6.2) does not change.

Problem (6.1)-(6.2) is linear and does not contain inequalities. There are virtually infinite ways of complicating this problem. Actually, optimization problems are a kind of Swiss-army knife in power system analysis and can be used for tackling a huge variety of power system issues. The scope of this chapter is not to enumerate all possible formulations but rather to provide the tools for solving an as general as possible class of optimization problems. With this aim, following sections focus only nonlinear programming (NLP) techniques and, among all possible NLP methods, only the reduced gradient method and the interior point methods are discussed.

6.2Optimal Power Flow Model

The very first formulations of the optimal power flow problem taking into account both power flow equations and economic dispatch were presented at the beginning of the sixties [48, 290] and further important developments were given in the following decade. At that time, there was no distinction between the solution method and the problem formulation. On the contrary, solution methods were suited to a specific problem formulation. For example, reduced gradient method in [80], Powell’s and Fletcher-Powell’s methods [265], Hessian method in [266], sequential programming [255], di erential injection method in [49], and linear programming [301]. By the middle of the seventies, the optimal power flow problem was a mature tool for assessing both power system security and economic dispatch [50, 120, 267, 309]. Since then, a huge variety of methods/models have been proposed so that survey papers have been necessary (e.g., [134, 300]).

In the last decade, power system restructuring has led to a renewed interest in mathematical programming, since it provides the adequate tools for tackling electricity markets [277]. From the programming point of view, in the last decade, the trend has been to take implementation details and problem formulation separated. This separation has been made possible by the

availability of mature software packages that provide a general purpose metalanguage for mathematical programming (e.g., GAMS [31] and AMPL [99]). In this chapter, modelling and implementation are also discussed separately.

The system model that is used throughout this chapter is a constrained nonlinear programming problem in the following general form:

where z Rnz , ϕ(z) is the objective function (ϕ(z) : Rnz → R), g(z) are the equality constraints (g(z) : Rnz → Rng ), and h(z) are the inequality constraints (h(z) : Rnz → Rnh ), and ng < nz . The latter condition allows (6.7) having (nz − ng ) degrees of freedom.1 The functions ϕ(z), g(z) and h(z) are assumed to be smooth, i.e., continuous and di erentiable at least two times for z Rnz .

According to the definitions given in Section 1.4 of Chapter 1, the vector z is formed by algebraic variables y and controllable parameters η. Thus, z = [yT , ηT ]T and nz = ny + nη .

In order to properly describe the solution methods for the problem (6.7), some definitions are required.

1.A point z Rnz is a local minimizer if g(z ) = 0 and h(z ) ≤ 0 and there exists an > 0 such that ϕ(z ) ≤ ϕ(z) x Rnz , with g(z) = 0 and h(z) ≤ 0 and |z − z | < .

2.A point zr Rnz is said to be a regular point of the constraints g(z) and h(z) if satisfies the conditions g(z) = 0 and h(z) ≤ 0 and if the gradients gzr and hzr are linearly independent.

The former definitions allows formulating the first-order KKT optimality conditions, as follows. If z is both a local minimizer of (6.7) and a regular point of the constraints g(z) and h(z), then there exist vectors ρ Rng and π Rnh , with π ≥ 0, such that:

The vectors ρ and π are called dual variables or multipliers. The first-order optimality conditions (6.8) can be conveniently expressed in terms of the Lagrangian function. The Lagrangian of the constrained problem (6.7) is:

1This definition is actually borrowed from mechanical engineering and is not used in mathematical programming.

The latter problem is somewhat easier to solve than (6.10)-(6.14). Analogously to the non-transformed problem, the conditions (6.18), (6.19) and (6.21) ensure the primal feasibility; the conditions (6.17) and ((6.22)) ensure the dual feasibility; and (6.20) are the complementarity conditions.

The concept of degenerated constraints completes this brief list of definitions. A constraint is degenerate if it is binding and its associated multiplier is null. For example, an inequality constraint is degenerate if hk = 0 and πk = 0, and is non-degenerate otherwise. In physical optimization problems and in particular in optimal power flow problems, binding constraints are generally non-degenerate. This observation eases the solution of the firstorder optimality conditions. Assuming non-degenerate binding constraints (6.10)-(6.14) become:

and the first-order optimality conditions (6.17)-(6.22) of the transformed problem (6.16) become:

Mathematical programming is a broad branch of mathematics and this section is not intended to provide a comprehensive treatise. The interested reader can find useful the following references [24, 51, 95, 96, 106, 175].